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Appl. Comput. Math., V.9, N.1, 2010, pp.

QUASILINEARIZATION APPROACH FOR SOLVING VOLTERRA’S

POPULATION MODEL

K. PARAND1, M. GHASEMI2, S. REZAZADEH2, A. PEIRAVI2, A. GHORBANPOUR2,

A. TAVAKOLI GOLPAYGANI3

Abstract. The study used the Quasilinearization method to solve Volterra’s model for popu-

lation growth of a species within a closed system is proposed. This model is a nonlinear integro-

diﬀerential where the integral term represents the eﬀect of toxin. First we convert this model

to a nonlinear ordinary diﬀerential equation, then approximate the solution of this equation by

treating the nonlinear terms as a perturbation about the linear ones. Finally we compare this

method with the other methods and come to the conclusion that the Quasilinearization method

gives excellent results.

Keywords: Volterra’s Population Model, Quasilinearization, Integro-diﬀerential, Barycentric.

AMS Subject Classiﬁcation:

1. Introduction

The Quasilinearization method was developed many years ago in the theory of linear pro-

gramming by Bellman and Kalaba [1, 5] as a generalization of the Newton Raphson method

[4, 11] to solve the systems of nonlinear ordinary and partial diﬀerential equations. Modern

developments and applications of the QLM method to diﬀerent ﬁelds are given in a monograph

[6]. This approach is applicable to a general nonlinear nth order ordinary or partial diﬀerential

equation in N-dimensional space, and to the complicated nonlinear two-point boundary condi-

tion [1, 5]. To reach convergence, the simple guess of the zeroth iteration being equal to zero or

to one of the boundary conditions, is usually enough [7]. This approach was applied for solving

nonlinear Calogero equation in a variable phase approach to quantum mechanics. The results

were compared with those of perturbation theory and the exact solutions. In this method iter-

ations yield rapid quadratic convergence and often monotonicity. In [8] the Quasilinearization

method was applied to solve diﬀerent nonlinear ordinary diﬀerential equations in physics, such

as Lane-Emden, Thomas-Fermi, Duﬃng and Blasius equations. They have shown this method

gives excellent results. These equations contain not only quadratic nonlinear terms but also

various other forms of nonlinearity of the higher derivatives. It was shown that a small number

of the QLM iterations yield fast convergence and uniformly stable numerical results.

The Volterra’s model for population growth of a species within a closed system is given in

[15, 12, 14] as

dp

d˜

t=ap −bp2−cp

˜

t

Z

0

p(x)dx, p(0) = p0,(1)

1Department of Computer Science , Shahid Beheshti University, Tehran, Iran

2Department of Applied Mathematics, Shahid Beheshti University, Tehran, Iran

3Department of Medical physics and Biomedical Engineering, Shiraz University of Medical Sciences,

Shiraz, Iran

Manuscript received 20 September 2008.

1

2 APPL. COMPUT. MATH., V.9, N.1, 2010

where a > 0 is the birth rate coeﬃcient, b > 0 is the crowding coeﬃcient, and c > 0 is the

toxicity coeﬃcient. The coeﬃcient cindicates the essential behavior of the population evolution

before its level falls to zero in the long term. p0is the initial population, and p=p(˜

t) denotes

the population at time ˜

t. This model is a ﬁrst-order integro-ordinary diﬀerential equation where

the term cp

˜

t

R

0

p(x)dx represents the eﬀect of toxin accumulation on the species. we apply scale

time and population by introducing the nondimensional variables

t=˜

tc

b, u =pb

a(2)

to produce the nondimentional problem

κdu

dt =u−u2−u

t

Z

0

u(x)dx, u(0) = u0,(3)

where u(t) is the scaled population of identical individuals at time t, and κ=c/(ab) is a

prescribed nondimensional parameter. The only equilibrium solution of (3) is the trivial solution

u(t) = 0 and the analytical solution [13]

u(t) = u0exp(1

κ

t

Z

0

[1 −u(τ)−

τ

Z

0

u(x)dx]dτ),

shows that u(t)>0 for all tif u0>0. In [13], the singular perturbation method for Volterra’s

population model is considered. It is shown in [13] that for the case κ¿1, where populations are

weakly sensitive to toxins, a rapid rise occurs along the logistic curve that will reach a peak and

then is followed by a slow exponential decay. And, for κlarge, where populations are strongly

sensitive to toxins, the solutions are proportional to sech2(t).In [15], the series solution method

and the decomposition method are implemented independently to (3) and to a related nonlinear

ordinary diﬀerential equation. Furthermore, the Pade approximations are used in the analysis

to capture the essential behavior of the populations u(t) of identical individuals. He compared

the approximation of umax and exact value of umax for diﬀerent κ.

The solution of (1) has been of considerable concern. Although a closed form solution has

been achieved in [14, 13], but it was formally shown that the closed form solution cannot lead to

any insight into the behavior of the population evolution [15]. Some approximate and numerical

solutions for Volterra’s population model have been reported. In reference [12], the successive

approximations method was oﬀered for the solution of (3), but was not implemented. In this case

the solution u(t) has a smaller amplitude compared to the amplitude of u(t) for the case κ¿1.

In [14] , several numerical algorithms namely the Euler method, the modiﬁed Euler method, the

classical fourth-order Runge-Kutta method and Runge-Kutta-Fehlberg method for the solution

of (3) are obtained. Moreover,a phase-plane analysis is implemented. In [14], the numerical

results are correlated to give insight on the problem and its solution without using perturbation

techniques. However, the performance of the traditional numerical techniques is well known

in that it provides grid points only, and in addition, it requires large amounts of calculations.

In [9, 10] applied spectral method to solve Volterra’s population on a semi-inﬁnite interval.

This approach is based on a Rational tau method. They obtained the operational matrices of

derivative and product of rational functions and reduced the solution of this problem to the

solution of algebraic equations systems.

As one more step in this direction, We use Quasilinearization approach for solving Volterra’s

population model and then we apply Baricentric Lagrange Interpolation with xj’s chebyshev

K. PARAND, M. GHASEMI, S. REZAZADEH, A. PEIRAVI, A. GHORBANPOUR, A. TAVAKOLI GOLPAYGANI: ...3

points of the ﬁrst kind. on the other hand, comparison of results obtained by this method and

others shows that this method provides more accurate and numerically stable solution than

those obtained by the other methods.

The paper is arranged as follows: in section 2 we describe the Quasilinearization method. In

this section the Volterra’s population model is considered. This equation is ﬁrst converted to

an equivalent nonlinear ordinary diﬀerential equation. Then we apply our method to solve this

ODE. In this section, the proposed method is used for the numerical examples, and a comparison

is made with the existing methods in the literature. Then, the results and numerical tables are

stated. The Finding of the study are summarized in the last section.

2. The Quasilinearization method (QLM)

The Quasilinearization method (QLM) is an application of the Newton-Raphson-Kantrovich

approximation method in function space [1]. This method is applied to solve a nonlinear nth

order ordinary or partial diﬀerential equation in N-dimensions as a limit of a sequence of linear

diﬀerential equations. The idea and advantage of the method is based on the fact that linear

equations can often be solved analytically or numerically using superposition principle while

there are no useful techniques for obtaining the general solution of a nonlinear equation in terms

of a ﬁnite set of particular solutions. As we discussed in the introduction, we limit ourselves here

to a nonlinear ordinary diﬀerential equation in one variable on the interval [0,b] which could be

inﬁnite:

L(n)y(x) = f(y(x), y(1)(x), ...., y(n−1)(x), x),(4)

with n boundary conditions

hi(y(0), y(1)(0), ..., y(n−1) (0)) = 0, i = 1, .., d (5)

and

hi(y(b), y(1)(b), ..., y(n−1)(b)) = 0, i =d+ 1, .., n, (6)

here L(n)is the linear nth order ordinary diﬀerential operator, fand h1, h2, ..., hnare the non-

linear functions of y(x) and its (n-1) derivatives are y(j)(x)j= 1, ..., n −1. The QLM

prescription [1, 5, 7] determines the (r+ 1)th iterative approximation yr+1(x) to the solution of

(4) as a solution of the linear diﬀerential equation:

L(n)yr+1(x) = f(yr(x), y(1)

r(x), ...., y(n−1)

r(x), x)+ (7)

+

n−1

X

j=0

(y(j)

r+1(x)−y(j)

r(x))fy(j)(yr(x), y(1)

r(x), ...., y(n−1)

r(x), x),

where y0

r(x) = yr(x), with linearized two-point boundary conditions

n−1

X

j=0

(y(j)

r+1(0) −y(j)

r(0))hky(j)(yr(0), y(1)

r(0), ...., y(n−1)

r(0),0) = 0, k = 1, .., d (8)

and

n−1

X

j=0

(y(j)

r+1(b)−y(j)

r(b))hky(j)(yr(b), y(1)

r(b), ...., y(n−1)

r(b), b) = 0, k =d+ 1, .., n. (9)

Here the functions fy(j)=∂f

∂yjare functional derivatives of the functionals

f(y(x), y(1)(x), ...., y(n−1)(x), x)

4 APPL. COMPUT. MATH., V.9, N.1, 2010

and

hk(y(x), y(1)(x), ...., y(n−1)(x), x)

respectively. The zeroth approximation y0(t) is chosen from mathematical or physical consid-

erations. Where δyr+1 (x) and ∆yr+1(x) are the diﬀerences between two subsequent iterations

and between the exact solution and the rth iteration, respectively:

δyr+1(x) = yr+1(x)−yr(x),

∆yr+1(x) = y(x)−yr(x).

In [1] proved k∆yr+1(x)k≤ kk∆yr(x)k2that showed the diﬀerence between the exact solution

and the rth iteration is decreasing quadratically and δyr+1(x) for an arbitrary l < r satisﬁed the

following inequality:

kδyr+1(x)k≤ (kkδyl(x)k)2r−l/k,

or for l= 0, they related the (n+ 1)th order result to the 1st iterate by

kδyn+1(x)k≤ (kkδy1(x)k)2n/k.

Therefore the convergence depends on the quantity q1=kky1−y0k.Where the zeroth

iteration y0(x) is chosen from physical and mathematical considerations.

3. Solving Volterra’s population model

3.1. Converting Volterra’s Population Model to a Nonlinear ODE Equation. In this

section we convert Volterra’s population model (3) to an equivalent nonlinear ordinary diﬀeren-

tial equation. Let

y(x) =

x

Z

0

u(t)dt. (10)

This leads to

y0(x) = u(x),(11)

y00(x) = u0(x).(12)

Inserting (10)-(12) into (1) yields the nonlinear diﬀerential equation

κy00 (x) = y0(x)−(y0(x))2−y(x)y0(x),(13)

with the initial conditions

y(0) = 0,

y0(0) = u0.

that are obtained by using (10) and (12) respectively.

K. PARAND, M. GHASEMI, S. REZAZADEH, A. PEIRAVI, A. GHORBANPOUR, A. TAVAKOLI GOLPAYGANI: ...5

3.2. Solution of Volterra’s Population Model. In the previous section we converted Volterra’s

Population model to a nonlinear ODE. In this section we apply the QLM procedure to solve

this model which is reduced to setting

κy00

n+1(x) = f(yn(x), y0

n(x)) + (yn+1(x)−yn(x))fyn+ (y0

n+1(x)−y0

n(x))fy0

n,

where f=y0(x)−(y0(x))2−y(x)y0(x). The (n+ 1)th iteration for volterra’s population model

has the following form:

κy00

n+1(x) = y0

n(x)(y0

n(x) + yn(x)−yn+1(x)) + y0

n+1(x)(1 −2y0

n(x)−yn(x)),

yn+1(0) = 0, y0

n+1(x) = 0.1,(14)

for κ= 0.02,0.04,0.1,0.2 and 0.5, where yn(x) is a previous iteration which is considered to be

a known function. We solve (14) numerically using Maple software and then apply Barycentric

Lagrange Interpolation to approximate yn+1(x) as the following [2, 3]:

yn+1(x) =

m

P

j=0

wj

(x−xj)fj

m

P

j=0

wj

(x−xj)

,

where xj’s are Chebyshev points of the ﬁrst kind that are given by:

xj= cos (2j+ 1)π

2m+ 2 , j = 0, .., m.

In this case, after cancelling factors independent of jwe have:

wj= (−1)jsin (2j+ 1)π

2m+ 2

and fj’s are the solutions of (14) in xj’s. The initial guess, satisfying the boundary conditions

at zero, was suitably made for diﬀerent values of κ.

3.3. Illustrative example. We apply the presented method in this paper to examine the math-

ematical structure of u(t). In particular, we seek to study the rapid growth along the logistic

curve that will reach a peak, then followed by the slow exponential decay where u(t)→0 as

t→ ∞. The mathematical behavior deﬁned in this way was introduced by [12] and justiﬁed by

[13] by using singular perturbation methods for the inner and outer solutions. Further, these

properties were also conﬁrmed by [14] upon using a phase plane analysis and [15] by using Pade

approximations. We apply the method presented in this paper and solved (3) where u0= 0.1

and κ= 0.02,0.04,0.1,0.2, and 0.5. In Table 1, we compared the results of QLM, method and

exact values of umax for κ= 0.02,0.04,0.1,0.2 and 0.5. In this table muis the minimum QLM

iteration number required in case that the absolute value of diﬀerences among the successive

iterations is less than (10−5).

6 APPL. COMPUT. MATH., V.9, N.1, 2010

Table 1. A comparison of the presented method(QLM) and the exact values for umax

κ muQLM Exact umax

0.02 5 0.92343412 0.92342717

0.04 4 0.87381153 0.87371998

0.1 7 0.76974153 0.76974149

0.2 3 0.65905030 0.65905038

0.5 8 0.48519041 0.48519030

In Fig.1 below is shown the results of QLM calculations for κ= 0.02,0.04,0.1,0.2 and 0.5.

0 1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k=0.02

k=0.04

k=0.1

k=0.2

k=0.5

Figure 1. The results of QLM calculations for κ= 0.02,0.04,0.1,0.2 and 0.5.

In addition, Fig.1 formally shows the rapid rise along the logistic curve followed by the slow ex-

ponential decay after reaching the maximum point.

In Table 2, the resulting values using the present method together with the results obtained

in [9, 10, 15] and the exact values are presented.

Table 2. A comparison of the method in [9, 10, 15] and the presented method with the exact values for umax.

κMethod in [9] Method in [10] Method in [15] QLM Exact umax

0.02 0.923327 0.923463 0.90383805 0.92343412 0.92342717

0.04 0.873605 0.873708 0.86124018 0.87381153 0.87371998

0.1 0.769623 0.769734 0.76511308 0.76974153 0.76974149

0.2 0.658872 0.659045 0.65791231 0.65905030 0.65905038

0.5 0.485076 0.485188 0.48528235 0.48519041 0.48519030

K. PARAND, M. GHASEMI, S. REZAZADEH, A. PEIRAVI, A. GHORBANPOUR, A. TAVAKOLI GOLPAYGANI: ...7

4. Concolusion

The Quasilinearization method(QLM)is very easy to apply and this method treats the non-

linear terms as perturbation about the linear ones and is not based,unlike perturbation theories,

on the existence of some kind of small parameters.

We used QLM for solving Volterra’s Population Model that in each stage,we solved yn(x) nu-

merically and applied Barycentric Lagrange Interpolation to approximate yn+1(x) with Cheby-

shev points. This interpolation is very good and accurate.

In the end we Compared this method with the other methods and found that the answers

were very accurate, numerically stable and converged fast.

We used an illustrative example to demonstrate the application of this method, and came to

satisfactory results.

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